Topological bands in one-dimensional periodic potentials

TL;DR

This paper investigates topological properties of quantum states in one-dimensional periodic potentials, revealing edge states, calculating Chern numbers, and linking particle pumping to topological invariants.

Contribution

It introduces a comprehensive analysis of topological invariants in 1D systems with shifted potentials, including exact Chern number calculations and their physical implications.

Findings

Edge states appear in energy gaps indicating non-trivial topology

Chern number of each band is exactly one in the continuous limit

Particle pumping is directly related to topological invariants

Abstract

We study the properties of the quantum states in the one-dimensional system with a shifted periodic potential in both the discrete model and the continuous model. With open boundary conditions, the edge states appear in the energy gaps which indicate non-trivial topological structures. The Chern numbers with respect to the Bloch vector and the potential shift angle are computed. In the limit of the continuous model, the Chern number of each band is exactly one. We demonstrate the particle number pumped by the adiabatically shift of the potential is directly related to the topological invariants.